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[RGC_man]

Ever clicked on that little link at the bottom of every page?

I like this software, and some of the message boards people start are fascinating.

http://www.thebagforum.com/

I never knew handbags could be such fun!

http://www.opticalinvestor.com/

This one was a disappointment. Was hoping it was full of generous millionaires.

http://forum.muppetcentral.com/

Yes, I do have too much time on my hands!

[rinselberg]

Picking up right where I stopped, this is a straightforward demonstration of conditional probabilities. We need only follow faithfully in the antique footsteps of one widely celebrated Thomas Bayes; to wit:

P(w|t) is the probability that you are a werewolf, given a positive test result. This is the probability that we need to arrive at - the answer to the puzzle. And here are the steps along the way ...

P(t|w) is the probability that you tested positive, given that you are a werewolf.

P(w) is the probability that you are a werewolf.

P(n) is the probability that you are not a werewolf. Either you are, or you aren't, and those are the only possibilities; therefore, we observe that ...

P(w) + P(n) = 1

... where 1 is the probability associated with a certainty. And that summation of the two mutually exclusive possibilities can be rearranged to yield ...

P(n) = 1.0 - P(w)

P(t) is the probability, a priori, of a positive test result; whether you are, in fact, a werewolf (accurate test result), or not a werewolf (false positive test result).

According to Bayes Theorem ...

P(w|t) = P(t|w) * P(w) / P(t)

Let's fill in the numbers that are straightforward from the "hypotheticals" ...

P(t|w) = 0.99 ... and P(w) = 0.01

Going back to the definition of P(n) as stated (above) and filling in the value of P(w), we have ...

P(n) = 1 - 0.01 ... which reduces to 0.99

But P(t) is not so straightforward. It is the sum of the probability of having a positive test result given that you are a werewolf, plus the probability of having a positive test result given that you are not a werewolf.

P(t) = P(t|w) * P(w) + P(t|n) * P(n)

This translates to ...

P(t) = 0.99 * 0.01 + 0.01 * 0.99

And that can be reduced to ...

P(t) = 2 * 0.01 * 0.99

Plugging all of these numbers back into "Bayes" (i.e. the equation given by Bayes Theorem; above) yields ...

P(w|t) = 0.99 * 0.01 / (2 * 0.01 * 0.99)

And that reduces to 1/2 or 0.5 ... just one chance in two that you are a werewolf. Even up. Fifty-fifty. Flip of a coin ... Since it's a given that there is only a one percent chance, a priori, that you are a werewolf, there is a 99 percent chance, a priori, that you are one of the unaffected poplulation. And it's just as likely that you are among the one percent of the unaffected population that is expected (statistically) to receive a false positive test result.

If your first thought was that there is a 99 percent chance that you're a werewolf, based on your positive test result, then you're no different than me. This might be called a "veridical paradox", because the answer seems counter-intuitive. Or as another OptiBoard member put it: "Bayes gives me a headache."

For another example of a veridical paradox, see Three Card Rinsel.

[rinselberg]

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